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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 236: fputs() expects parameter 1 to be resource, bool given
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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 236: fputs() expects parameter 1 to be resource, bool given
[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 237: fclose() expects parameter 1 to be resource, bool given
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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 236: fputs() expects parameter 1 to be resource, bool given
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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 236: fputs() expects parameter 1 to be resource, bool given
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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 236: fputs() expects parameter 1 to be resource, bool given
[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 237: fclose() expects parameter 1 to be resource, bool given
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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 236: fputs() expects parameter 1 to be resource, bool given
[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 237: fclose() expects parameter 1 to be resource, bool given
[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 280: copy(dc19d06345f493857c38e85b44f01884.png): failed to open stream: No such file or directory
[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 281: chmod(): No such file or directory
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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 236: fputs() expects parameter 1 to be resource, bool given
[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 237: fclose() expects parameter 1 to be resource, bool given
[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 280: copy(dc19d06345f493857c38e85b44f01884.png): failed to open stream: No such file or directory
[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 281: chmod(): No such file or directory
[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 295: chdir(): No such file or directory (errno 2)
[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 297: unlink([ROOT]/latexrender//tmp/dc19d06345f493857c38e85b44f01884.tex): No such file or directory
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[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 235: fopen([ROOT]/latexrender//tmp/dc19d06345f493857c38e85b44f01884.tex): failed to open stream: No such file or directory
[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 236: fputs() expects parameter 1 to be resource, bool given
[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 237: fclose() expects parameter 1 to be resource, bool given
[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 280: copy(dc19d06345f493857c38e85b44f01884.png): failed to open stream: No such file or directory
[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 281: chmod(): No such file or directory
[phpBB Debug] PHP Warning: in file [ROOT]/latexrender/class.latexrender.php on line 295: chdir(): No such file or directory (errno 2)
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forum.virtualchallengemeets.com • View topic - RMO 2001

RMO 2001

Pose Olympiad-level questions to the monkeys in the peanut #.

RMO 2001

Postby zefuri » Thu Apr 19, 2007 5:49 pm

Find all primes [unparseable or potentially dangerous latex formula] such that [unparseable or potentially dangerous latex formula]
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Postby makashi » Wed May 09, 2007 6:33 pm

For primes greater than three, [unparseable or potentially dangerous latex formula] or [unparseable or potentially dangerous latex formula]. But in [unparseable or potentially dangerous latex formula],
[unparseable or potentially dangerous latex formula]

The remaining cases consist of at least one prime being either two or three. But note that there is only one case in which [unparseable or potentially dangerous latex formula] or [unparseable or potentially dangerous latex formula] in which the both sides of the equation are positive ([unparseable or potentially dangerous latex formula] which does not have equality). Hence, the only possibilities remain with [unparseable or potentially dangerous latex formula] or [unparseable or potentially dangerous latex formula].

Suppose [unparseable or potentially dangerous latex formula]. We have already dealt with the cases in which [unparseable or potentially dangerous latex formula] or [unparseable or potentially dangerous latex formula]. For [unparseable or potentially dangerous latex formula], [unparseable or potentially dangerous latex formula], so there are no solutions with [unparseable or potentially dangerous latex formula].

Suppose [unparseable or potentially dangerous latex formula]. There is one solution with [unparseable or potentially dangerous latex formula]. For [unparseable or potentially dangerous latex formula], [unparseable or potentially dangerous latex formula], so there are no solutions with [unparseable or potentially dangerous latex formula].

Therefore, the only solution is [unparseable or potentially dangerous latex formula].
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